Thanks for the work your doing here. But I’ve not been able to figure out the introduction of j into into the proof, even after reading the explanation you gave in the comment section.Please can you like explain again how your going about your proof, like how you know where your headed….

Hi, no problem, comment away! (Though if I get busy, responses could start to lag.)
We have

But we know for any number that it is between and for exactly one integer . But since we know that integer is somewhere between and , which means it is for some integer . So, then we have that is between and . Does that make sense? Basically, we are just saying that we know for some integer , but then it is more convenient to write it as since that will help us out later.

Not sure if this actually clears up your question. Let me know if not.